---
tag: 4W27TKGQWR
kind: concept
name: Buekenhout geometry
msc: ["51E24", 51A05]
sources:
  - url: https://en.wikipedia.org/wiki/Buekenhout_geometry
same_as:
  wikidata: Q4985363
  wikipedia: Buekenhout geometry
---

## Summary

A Buekenhout geometry, or diagram geometry, is a set of objects of several types with a symmetric incidence relation.
Every flag of pairwise incident objects extends to a chamber, which contains one object of each type.
It generalizes [[projective spaces|ZT9A6JK8QF]], [[generalized polygons|PCT2HY9AAZ]] and Tits buildings.
Francis Buekenhout introduced a diagram that records, for each pair of types, which rank $2$ geometries appear.
These diagrams are used to describe and classify geometries, including geometries for sporadic simple groups.

## Explanations

### L1 human

A projective space has points, lines, planes and so on, and a point lies on a line, a line lies in a plane, and so forth.
A Buekenhout geometry keeps only this pattern: objects of different types and a relation saying when two objects are incident.
Each pair of types gives a small two-type geometry, such as a projective plane or a generalized quadrangle.
A diagram with one node per type and labelled edges records these pairs, much like a Dynkin diagram.